Download Axion-like particle production in a laser

Axion-like particle production in a laser-induced dynamical spacetime
M. A. Wadud,1, 2 B. King,3 R. Bingham,4, 5 and G. Gregori1
1
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
School of Physics & Astronomy, 116 Church Street S.E., Minneapolis, MN 55455, USA
3
School of Computing, Electronics and Mathematics, Plymouth University, Devon, PL4 8AA, UK
4
University of Strathclyde, Glasgow, G4 0NG, UK
5
Rutherford Appleton Laboratory, Chilton, Didcot OX11 0QX, UK
(Dated: December 23, 2016)
arXiv:1612.07743v1 [hep-ph] 22 Dec 2016
2
We consider the dynamics of a charged particle (e.g., an electron) oscillating in a laser field
in flat spacetime and describe it in terms of the variable mass metric. By applying Einstein’s
equivalence principle, we show that, after representing the electron motion in a time-dependent
manner, the variable mass metric takes the form of the Friedmann-Lemaı̂tre-Robertson-Walker
metric. We quantize a massive complex scalar or pseudo-scalar field in this spacetime and derive
the production rate of electrically neutral, spinless particle-antiparticle pairs. We show that this
approach can provide an alternative experimental method to axion searches.
It is well known that particle-production phenomena
can occur in a curved or dynamic spacetime [1]. For
example, thermal radiation can arise from particle production near the event horizon of a black hole, an effect
commonly known as the Hawking radiation [2, 3]. This
is a quite general fact, not confined to black holes. As
hypothesized by Davies, Unruh and Fulling [4–6], an observer in a uniformly accelerated frame experiences the
surrounding vacuum as filled with thermal radiation with
temperature TDU = ~a/2πkB c = 4.05 × 10−23a K, where
a is the acceleration (in cm/s2 ) and kB is the Boltzmann constant. The expansion of the universe also gives
rise to a curved metric called the Friedmann-Lemaı̂treRobertson-Walker (FLRW) metric: ds2 = dt2 −h2 (t)dx2 .
Here h(t) is the scale factor which quantifies the relative
expansion of the universe. In the FLRW metric, particles
are spontaneously produced as a result of the expansion
of the universe [7–10]. Of particular interest is the inflationary period, from 10−36 s until 10−32 s after the big
bang. During this time, it is thought that the universe
expanded exponentially, and spacetime was highly curved
and dynamic. Understanding particle production during
inflation [11–13] may help answer major questions like
why the universe today is isotropic and flat, and why
there is more matter than antimatter [14, 15].
The latter is an example of a spontaneously broken
symmetry that may require the existence of particles
beyond the standard model. The axion is one of such
particles, a pseudo Nambu-Goldstone boson which arises
from the spontaneously broken Peccei-Quinn symmetry
[16]. Both astrophysical bounds from stars and galaxies
[17, 18] as well as laboratory searches [19, 20] have provided limits for the mass and coupling constants of these
hypothetical particles. While experimental searches so
far have not yet identified an axion candidate, the parameter space left to explore is still large and there is a
need of more sensitive probes before the axion existence
can be confidently ruled out.
Recent advancements in ultra-high intensity lasers
[21] have stirred interest in the possibility of detecting both the Schwinger effect and dynamic spacetime
phenomena [22–25]. Projects under development include the European Extreme Light Infrastructure [26],
which will provide radiation beams of intensities exceeding 1024 W/cm2 ; the X-ray free electron lasers (XFEL)
based at DESY Hamburg, and the LCLS (Linac Coherent
Light Source) facility at SLAC, where highly tunable xray pulses with narrow bandwidth and high intensity are
already available. Here we propose a mechanism for electrically neutral particle-antiparticle pair production in a
laser field, whereby the variation of the metric around a
charged particle oscillating in the laser field gives rise to
spontaneous particle-antiparticle pair production.
We start with the Lagrangian density of a free, massive, minimally-coupled complex scalar or pseudo-scalar
field φ(x) under the FLRW metric gµν = h2 (η)ηµν , where
ηµν = diag(1, −1, −1, −1) is the Minkowski metric and
h(η) is the scale factor [27]:
p
L = − det gµν g µν (∂µ φ∗ )(∂ν φ) − m2 φ∗ φ . (1)
Here x0 = η is the conformal time (not to be confused
with the Minkowski metric ηµν ) which is related to the
physical time t by dt/dη = h(η). Natural units (~ = c =
ǫ0 = 1) are used in all the derivations, and conversion to
physical units will be explicitly mentioned.
The equation of motion of the field
R is the one that extremises the action functional S = d4 xL. The extremisation condition is equivalent to Euler-Lagrange equation, giving for our case the Klein-Gordon equation. The
next step consists in the procedure of canonical quantisation of the field φ and provide a framework for particles to be created and annihilated. In FLRW spacetime,
however, the vacuum states at different times, |0iη0 and
|0iη1 are different, and a notion of particle and antiparticle that is consistent at all times is unattainable. To
circumvent the ambiguity about the vacuum state, we
first assume the existence of a preferred particle model
that provides time-independent creation/annihilation operators from which we can construct a reference vacuum
state. Such conditions are fulfilled, for example, when
looking at the solution of the Klein-Gordon equation at
2
asymptotic times (η → ±∞) [28–30]. These are used to
define time-dependent creation and annihilation operators, related to the asymptotic ones by Bogoliubov transformation. The procedure outlined above corresponds to
ω̇k (η)
dNk
(η) =
dη
2ωk (η)
Z
η
η0
dη ′
ω̇k (η ′ )
[1 + 2Nk (η ′ )] cos[2Θk (η) − 2Θk (η ′ )],
ωk (η ′ )
where Nk is the time-dependent number of pairs of spatial mode k,
ωk2 = k2 + m2 h2 −
ḧ
,
h
the kinetic approach to quantum field theory, leading to
the so called quantum Vlasov equation [29, 31]. We obtain (see e.g., [30]),
(3)
with the dot notation representing differentiation by η
(i.e. ḧ = d2 h/dη 2 ) and
Z η
dη ′ ωk (η ′ ).
(4)
Θk (η) =
The time η0 is defined such that Nk (η0 ) = 0. The quantum Vlasov equation, eq. (2), is formally similar to the
one obtained by Kluger et al. [29] for bosonic pair production in flat spacetime under an oscillating electron
field. However, in our case, it has been specialized such
that there is no explicit presence of an electric field and
the spacetime is more generally defined by the FLRW
metric. This is in fact generally the case for field theories
in background fields. Since the particle number operator does not, in general, commute with the interaction
Hamiltonian, one must be cautious interpreting results
at intermediate times. Different particle number definitions that coincide at asymptotic times, may disagree
by orders of magnitude at intermediate times (this phenomenon has been recently studied using the superadiabatic basis to analyse the Schwinger effect [32]). We note
the quantum Vlasov equation’s non-Markovian character
[33]: the term 1 + 2Nk (η ′ ) in the integral means that the
equation is non-local in time, i.e., the production rate of
pairs is dependent on the history of the system.
Having obtained the particle production rate in an expanding spacetime, we now describe the dynamics of
a particle in a laser field with an alternative metric
that, as we shall see, bears many resemblances with the
FLRW metric. In our approach we do not quantise the
laser electromagnetic field nor the metric, which will be
treated classically as existing in the background. Following closely the derivation by Crowley et al. [34, 35], we
consider the dynamics of a free particle of mass m0 under the variable mass metric [36]. The name of the metric derives from the appearance of the “variable mass”
hm0 in the place of the rest mass m0 in dynamical equations that are similar to flat spacetime equations. We
have gµν = h2 (x)ηµν , where ηµν is the Minkowski metric and h(x) is a spatial field. In general relativity, the
(2)
dynamics of a free particle of mass m0 in a spacetime
with metric gµν is determined by its Lagrangian [37]
L = −(gµν v µ v ν )1/2 m0 , where v µ = dxµ /dx0 is the 4velocity. The Lagrangian for the variable mass metric is
thus [34], L = −hm0 /γ, where γ = (1 − v 2 )−1/2 . The
canonical 3-momentum is given by p = ∂L/∂v = γhm0 v,
and the Hamiltonian is then H = p·v−L = γhm0 . Using
Hamilton’s equations then obtain:
H = (p2 + h2 m20 )1/2 ,
(5)
and,
a=−
1 ∂ ln h
γ 2 ∂x
(6)
where we have used the fact that the Hamiltonian has no
explicit time dependence.
Let us now consider the dynamics of a charged particle,
with charge q and mass m0 , oscillating, with frequency ν,
in a laser field with vector potential A and null scalar potential (Coulomb radiation gauge) in flat spacetime, ηµν .
The Lagrangian for this particle is L = −m0 /γ + qv · A,
which gives us the canonical momentum p = γm0 v + qA
and the Hamiltonian H = [(p − qA)2 + m20 ]1/2 . We can
decompose the momentum into parallel and perpendicular components with respect to A, that is p = pk + p⊥ .
Thus if v0 is the velocity of the particle due to the
influence of the laser field, and any remaining components are sufficiently small, we approximately have
(pk − qA)2 ≈ γ02 m20 v02 , with γ0 = (1 − v02 )−1/2 . We notice
that |p| ∼ |p⊥ | ≈ γ1 γ0 m0 v1 , where γ1 = (1 − v12 )−1/2 is
calculated with respect to a particle velocity v1 which is
not associated to the motions induced by the laser field
(that is, v1 = v − v0 , and v1 ⊥ v0 ). This holds under
the condition that either v0 ≪ 1 or v1 ≪ 1. Hence,
H = (p2 + γ02 m20 )1/2 ,
(7)
and
a=−
1 ∂ ln γ0
.
γ12 ∂x
(8)
We immediately notice that, if we make the substitutions
γ0 → h and γ1 → γ, these last two equations are the same
as (5) and (6). Einstein’s equivalence principle asserts
that an observer cannot distinguish between his frame’s
3
acceleration in flat spacetime and a metric field whose
geodesic has equal acceleration, i.e., the physics is the
same in both cases. Hence, the dynamics of the charged
particle oscillating in the laser field in flat spacetime may
be equivalently described by the variable mass metric
Hamiltonian of a free particle.
field, we make the approximation [39]
The idea that electromagnetic acceleration can give
rise to dynamics that can be equivalently described by
the variable mass metric will now be used to represent h in a time-dependent functional form so that it
becomes equivalent to the scale factor of the FLRW
metric. This key result will allow us to employ the
field quantisation formalism developed earlier and obtain
particle-antiparticle pair production rate with the quantum FLRW-Vlasov equation.
where we have assumed that Nk (η0 ) = 0. The integrand
is symmetric with respect to the exchange η ′ ↔ η ′′ , which
means it is symmetric about the line η ′ = η ′′ . Hence [40]
We take a laser pulse of frequency ν, four-wavevector
κ, phase ϕ = κ · x, duration τ and intensity parameter
ξ = qE/m0 ν to be represented at the focus by a vector
potential
ϕ 2
A = m0 ξ exp −
cos ϕ ẑ,
(9)
Φ
where Φ = ντ and ẑ is the unit vector in the z-direction.
We limit the analysis to non-relativistic electron motion
(γ0 ≈ 1), by specifying that ξ ≪ 1. Assuming the particle
begins at the origin with zero momentum in the infinite
past, the velocity component in the field direction is ẋ ·
ẑ = qA/m0 , which gives:
h = 1 − v02
−1/2
−1/2
q 2 A2
≈1+
= 1 − (qA/m0 )2
.
2m20
We see that h ≥ 1, meaning that space expands when the
electric field is non-zero. This can be interpreted as the
result of the increased energy density of free space due
to the presence of an electric field. In other terms, the
electron acquires an effective mass meff = hm0 . The idea
of an effective mass to describe the motion of electrons in
intense laser beams is not new, and it is associated with
the frequency shift of the radiation emitted by a particle
in an intense electromagnetic field [38].
With this time-dependent form of h, the variable mass
metric becomes equivalent to the FLRW metric. We can
thus use the quantum field formalism developed earlier
to estimate the particle production by integrating the
FLRW quantum Vlasov equation (2). In doing so, we
note that the field mass m appearing in equation (3) for
the frequency ωk is not necessarily the same as the mass
m0 of the oscillating particle. We assume that the test
particle being accelerated by the laser field has mass m0
and charge q while the particles that are being produced
as a result of the transformed metric have mass m and
no charge.
Next, in the low density regime, that is, when the laser
electric field is much smaller than the Schwinger’s critical
1
Nk (η0 , η) ≈
2
Z
η
η0
Z
η ′′
η0
dη ′′ dη ′
ω̇k (η ′ ) ω̇k (η ′′ )
×
ωk (η ′ ) ωk (η ′′ )
cos[2Θk (η ′ ) − 2Θk (η ′′ )],
2
Z
1 η ′ ω̇k (η ′ )
′ exp[2iΘk (η )] ,
Nk (η0 , η) ≈ dη
′
4 η0
ωk (η )
(10)
where we have used the fact that the antisymmetry of
the factor sin{2Θk (η ′ ) − 2Θk (η ′′ )} with respect to the
exchange η ′ ↔ η ′′ has null contribution to the integral
[39, 40].
From (3) we then have:
...
h h − ḣḧ
ωk ω̇k =m hḣ −
2h2
2
(11)
where ωk2 (η) ≈ k2 + m2 . The change in the metric perturbation depends only on the external field, so we have
h = h(ϕ), but on the other hand wish to integrate over
the conformal time, η. In general, the dependency η(ϕ)
can be complicated, but for a plane-wave background
we can write d/dη = Ω−1 d/dϕ where Ω = κ · p/m is
the particle energy parameter [41]. As previously mentioned, of experimental relevance are the asymptotic values of observables for times long after the laser pulse
has passed through the seed electrons [32]. For this reason, we integrate to finite phases, and in the final calculated observables, take the asymptotic limit. In this vein,
Nk (−ΩR, ΩR) ≈ 41 |Ik (R)|2 where:
#
...
Z R"
h h − ḣḧ 2iΘk
1
2
e
dϕ. (12)
m hḣ −
Ik (R) = 2
ωk Ω −R
2h2
Let us then define Nk = limR→∞ Nk (−ΩR, ΩR). By
assuming the hierarchy m0 ≫ m0 ξ ≫ Ω ≥ m and expanding to lowest order in Φ−1 , terms in pre-exponents
of order O(m2 ξ 4 Ω) and O(ξ 4 Ω3 ), O(m2 ξ 2 Ω/Φ2 ) were neglected in the integration. The leading-order terms were
then:
πξ 4 Φ2 (m2 + 2 Ω2 )2 h − Φ22 (ωk −Ω)2 − Φ22 (ωk +Ω)2 i
Nk ≈
+e Ω
e Ω
27 ωk4
Integrating over all modes k gives the total particle density:
Z
π Φ2 ξ 4 (m2 + 2 ν 2 )2
N = d3 k Nk (η) =
I2 (m, ν, τ ),
27
m
(13)
4
I2 (m, ν, τ ) =
Z
0
∞
i
dy y 2 h −τ 2 (ωk −ν)2
−τ 2 (ωk +ν)2
,
e
+
e
(y 2 + 1)2
(recalling Φ = ντ ), ωk2 = m2 (1+y 2 ) and we have assumed
the particle starts at rest, so Ω = ν. Using the Laplace
method for the case Φ ≫ 1 and a perturbative expansion
for when mτ ≪ 1, we find that if m ≪ ν:
 2 4 4
π ξ ν
2


Φ2 e−Φ , mτ ≪ 1
 7
2 m
(14)
N≈
3/2 4 4


π ξ ν τ ,
Φ
≫
1
25
(we recall that Φ ≫ 1 is still assumed in both of these
expressions).
In the short-pulse case, the produced axion density
tends to the familiar N ∼ 1/m. The dependency of axion
production on the axion mass is given in Fig. 1.
mN/ν4
mN/ν4
10-5
10-4
10-6
10-5
10-6
10-7
10-7
10-8
m
-4
10
0.001
0.01
0.1 ν
10-8
m
0.2
0.5
1
2
5 ν
FIG. 1. Mass dependency of mN/ν 4 for ξ = 0.1. Shaded
regions indicate axion masses ruled out by the calculation
in [42]. Left: as Φ is reduced, I3 becomes flat, indicating a
dependency 1/m. Right: As Φ is increased, the heaviest axion
mass to which the method is sensitive, decreases.
A question which immediately arises is how this mechanism of axion-like particle production compares, for ex-
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